1. Field of the Invention
The present invention relates to a parametric analyzing method for calculating the light intensity in a photolithography process used in a semiconductor fabrication apparatus.
2. Description of the Related Art
A photolithography process is one of the most fundamental and important fabrication processes for forming a desired pattern on a semiconductor substrate as a fabrication process for semiconductor devices such as integrated circuits.
A light exposing device used in the photolithography process has an optical system shown in FIG. 6. The optical system comprises a light source 61, a collimation lens 62, a mask 63, a condenser lens 64, and a photoresist 66. The collimated lens 62 converts light from the light source 61 into collimated light. The mask 63 has a predetermined mask pattern 63a. The condenser lens 64 forms an image of the mask pattern 63a. The photoresist 66 is disposed on a wafer 65 as an object of the photolithography process. Light emitted from the light source 61 is collimated by the collimation lens 62. The collimated light illuminates the mask 63. The light that has transmitted the mask pattern 63a on the mask 63 is condensed by the condenser lens 64. The condensed light reaches the photoresist 66 on the wafer 65 and forms an image of the mask pattern 63a. The light intensity distribution on the wafer 65 is a major factor that largely affects the resolution of the pattern, the reproducibility, and so forth.
Thus, when the light intensity distribution is analyzed, various parameters in the photolithography process can be optimized.
The light intensity distribution can be analyzed by a method using Hopkins' formula that has been disclosed in a periodical titled "Optical Technology Contact (translated title)", No. 28, 1990, pp. 165-175. According to the method for calculating light intensity using Hopkins' formula, the light intensity distribution on a wafer is given by Expression (1). EQU I(u, v)=.intg..intg..intg..intg.T(p, q; p', q')t(p, q)t* (p', q') exp[-i2.pi.((p-p')u+(q-q')v)]dp dp' dq dq' (1)
where t is the complex amplitude transmissivity of the mask; t* is the complex conjugate of t; t is the Fourier transform of t; T is the transmission cross-coefficient (TCC); P=lfx and q=mfy (l and m are integers, f.sub.x and f.sub.y are periodic frequencies in the x direction and y direction of the mask pattern, respectively).
The transmission cross-coefficient (TCC) in Expression (1) is given by Expression (2). EQU T(p, q; p', q')=.intg..intg.S(r, s)P(p+r,q+s)P*(p'+r,q'+s) exp[-i.pi..delta.((p+r).sup.2 +(q+s).sup.2 -(p'+r).sup.2 -(q'+s).sup.2)]dr ds (2)
where S is an effective light source; P is a pupil function; P* is the complex conjugate of P; and .delta. is defocus.
The coordinates (u, v) on the wafer in Expression (1) and the defocus .delta. in Expression (2) have been normalized. The coordinates (u, v) and the defocus .delta. have the following relation with the real coordinates (U, V) and the real defocus .DELTA.. EQU U=u.lambda./NA (3-1) EQU V=v.lambda./NA (3-2) EQU .DELTA.=.delta..lambda./(NA).sup.2 (3-3)
where .lambda. is the wavelength of light; and NA is the numerical aperture of the condenser lens.
To calculate the light intensity distribution I (u, v) with Expressions (1) and (2), assuming that l and m are integers and that p=lfx and q=mfy, from the cut-off of the pupil (that is a condition of which S(r, s) P(p+r, q+s) P*(p'+r, q'+s) is not 0), Expressions (4-1) and (4-2) are satisfied. Thus, assuming that the minimum integers l and m that satisfy Expressions (4-1) and (4-2) are the maximum degrees l.sub.max and m.sub.max of Fourier transform, Expressions (1) and (2) are calculated for up to the maximum degrees l.sub.max and m.sub.max. EQU (1+.sigma.)/f.sub.x .ltoreq.1 (4-1) EQU (1+.sigma.)/f.sub.y .ltoreq.m (4-2)
where .sigma. is a coherence factor; and f.sub.x and f.sub.y are the periodic frequencies in the x direction and y direction of the mask pattern.
The above-described calculating procedure is shown with a flow chart in FIG. 5. First, initial conditions such as the coherence factor .sigma., the mask conditions (for example, the transmissivity of the mask and the position of the pattern), and the optical conditions (for example, the light source and pupil) are set (at step 51). Next, with Expression (4), the maximum degrees l.sub.max and m.sub.max are calculated (at step 52). With Expression (2), the transmission cross-coefficients are calculated for up to the maximum degrees l.sub.max and m.sub.max and the calculated results are tabulated as a table (at step 53). The Fourier transforms of the transmissivity t of the mask (hereinafter referred to as Fourier transforms of the mask) are calculated for up to the maximum degrees l.sub.max and m.sub.max and then the calculated results are tabulated as a table (at step 54). With the tables generated at step 53 and 54 and Expression (1), the light intensity distribution I (u, v) on the wafer is calculated (at step 55). It is determined whether or not the light intensity distributions I (u, v) have been calculated for all the conditions (at step 56). When the determined result at step 56 is No, the conditions of the coherence factor .sigma., the transmissivity of the mask, the position of the pattern, the light source, the pupil, and so forth are varied for the conditions and steps 52 to 56 are performed. Until the light intensity distributions I are calculated for all the conditions, steps 52 to 56 are repeated.
In the above-described conventional calculating method for the light intensity distribution, the transmission cross-coefficients and the Fourier transforms of the mask are calculated for all the conditions. Thus, it takes a very long time to complete such calculations.